Behavior of the Trinomial Arcs

In this paper, we deal with the family I(p, k, r, n) of trinomial arcs defined as the set of roots of the trinomial equation zn = αzk + (1−α), with z = ρ eiθ is a complex number, α is a real number between 0 and 1 and k is an integer such that k = (2p + 1)n/(2r + 1), where n, p and r are three integers satisfying some conditions. These arcs I(p, k, r, n) are continuous arcs inside the unit disk, expressed in polar coordinates (ρ, θ). The question is to prove that ρ changes monotonically with respect to θ and that ρ (θ) is a decreasing function, for each trinomial arc I(p, k, r, n). Mathematics Subject Classification: 03F65, 12D10, 14H45, 26C10, 30C15, 65H05